phi-FEM: a finite element method on domains defined by level-sets

  • Michel, Duprez (Inria)
  • Alexei, Lozinski (Université de Besançon)
  • Vanessa, Lleras (Université de Montpelier)

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We present a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we search the approximation to the solution as a product of a finite element function with the given level-set function, also approximated by finite elements. The imposition of Neumann boundary conditions is less straightforward and requires the introduction of auxiliary variables near the boundary. Unlike other recent fictitious domain-type methods (XFEM, CutFEM), our approach does not need any non-standard numerical integration, neither on the cut mesh elements nor on the actual boundary. We shall present the proofs of optimal convergence of our methods on the example of Poisson equation using Lagrange finite elements of any order. We will also give numerical tests illustrating the optimal convergence of our methods and discuss the conditionning of resulting linear systems and the robustness with respect to the geometry. We have also highlight the flexibility and efficiency of our method on elastic and dynamic problems. And more recently, in \cite{stokes}, we propose a phi-FEM formulation to solve particulate flows and Stokes equations.